Chapter P: Preliminaries

Fall 2015

Department of MathematicsHong Kong Baptist University

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History

Calculus was developed in the 17th century to study importantscientific and mathematical problems raised in physical science.For instance, it can be used to find

(1) the tangent line to a curve at a given point;

(2) the length of a curve, area of a region, and volume of a solid;

(3) the minima, maxima of quantities, e.g., the distance betweenthe earth and the sun;

(4) and many more ...

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Preliminaries

The preliminary chapter reviews the most important thingsthat you should know before beginning calculus.

Depending on your pre-calculus background, you may or maynot be familiar with the topics in the preliminaries chapter.

(1) If you are, you may want to skim over these materials torefresh your understanding of the terms used;

(2) If not, you should study this chapter in detail.

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§P.1 Real Numbers and the Real Line

Real numbers are numbers that can be expressed asdecimals, for example,

5 = 5.00000...

4

3= 1.33333...

√2 = 1.41421...

π = 3.14159...

e = 2.71828...

The real numbers can be represented geometrically as pointson the real line, denoted by R.

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Some important special subsets of real numbers are:

(i) natural numbers (or positive integers): the numbers 1, 2,3, 4, 5, ......

(ii) integers: the numbers 0, ±1, ±2, ±3, ±4, ±5, ......

(iii) rational numbers: the numbers that can be expressed in theform of a fraction m/n, where m and n are integers and n 6= 0.

(iv) irrational numbers: the real numbers that are not rational.

Example:√

2 is an irrational number. Why?

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Intervals

Let a and b be two real numbers and a < b. Then

(i) (a, b): the open interval from a to b, consisting of all realnumbers x satisfying a < x < b.

(ii) [a, b]: the closed interval from a to b, consisting of all realnumbers x satisfying a ≤ x ≤ b.

(iii) [a, b): the half-open interval from a to b, consisting of allreal numbers x satisfying a ≤ x < b.

(iv) (a, b]: the half-open interval from a to b, consisting of allreal numbers x satisfying a < x ≤ b.

Remark: Note that the whole real line is also an interval, denotedby R = (−∞,∞). The symbol ∞ is called “infinity”.

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Absolute Value

The absolute value, or magnitude, of a number x , denoted by|x |, is defined by the formula

|x | =

{x , if x ≥ 0−x , if x < 0.

Properties of absolute values include:

1. | − a| = |a|.

2. |ab| = |a| · |b| and |a/b| = |a|/|b|.

3. |a± b| ≤ |a|+ |b|.

4. If D is a positive number, then

|x − a| < D ⇐⇒ a− D < x < a + D.|x − a| > D ⇐⇒ either x < a− D or x > a + D.

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§P.2 Cartesian Coordinates in the Plane

The positions of all points in a plane can be measured withrespect to two perpendicular real lines (referred to as x-axisand y-axis) in the plane intersecting at the 0-point of each.

The point of intersection of the x-axis and the y -axis is calledthe origin and is often denoted by the letter O.

For a point P(a, b), we call a the x-coordinate of P, and bthe y-coordinate of P. The ordered pair (a, b) is called theCartesian coordinates of the point P.

The coordinate axes divide the plane into four regions calledquadrants, numbered from I to IV.

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(a) x2 + y 2 = 4, and (b) x2 + y 2 ≤ 4

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Straight Lines

Given two points P1(x1, y1) and P2(x2, y2) in the plane, thereis a unique straight line passing through them both. We callthe line L = P1P2.

For any nonvertical line L, we define the slope of the line is

m =∆y

∆x=

y2 − y1x2 − x1

= tan(φ),

where the angle φ is the inclination of the line L. We let0o ≤ φ < 180o (or 0 ≤ φ < π).

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Equations of Straight Lines

The point-slope equation of the line that passes through thepoint (x1, y1) and has slope m is

y = m(x − x1) + y1.

The point-point equation of the line that passes through thepoint (x1, y1) and (x2, y2) is

y =y2 − y1x2 − x1

(x − x1) + y1.

The horizontal line passing through the point (x1, y1) is

y = y1.

The vertical line passing through the point (x1, y1) is

x = x1.

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Equations of Straight Lines

The point-slope equation of the line that passes through thepoint (x1, y1) and has slope m is

y = m(x − x1) + y1.

The point-point equation of the line that passes through thepoint (x1, y1) and (x2, y2) is

y =y2 − y1x2 − x1

(x − x1) + y1.

The horizontal line passing through the point (x1, y1) is

y = y1.

The vertical line passing through the point (x1, y1) is

x = x1.

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Equations of Straight Lines

The point-slope equation of the line that passes through thepoint (x1, y1) and has slope m is

y = m(x − x1) + y1.

The point-point equation of the line that passes through thepoint (x1, y1) and (x2, y2) is

y =y2 − y1x2 − x1

(x − x1) + y1.

The horizontal line passing through the point (x1, y1) is

y = y1.

The vertical line passing through the point (x1, y1) is

x = x1.

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For a nonhorizontal and nonvertical line, let a be the x-interceptand b be the y-intercept. Then

The slope–y-intercept equation of the line with slope m andy -intercept b is

y = mx + b.

The slope–x-intercept equation of the line with slope m andx-intercept a is

y = m(x − a).

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For a nonhorizontal and nonvertical line, let a be the x-interceptand b be the y-intercept. Then

The slope–y-intercept equation of the line with slope m andy -intercept b is

y = mx + b.

The slope–x-intercept equation of the line with slope m andx-intercept a is

y = m(x − a).

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Example 1:

Find the slope and the two intercepts of the line with equation8x + 5y = 20.

Solution: Solving the equation for y , we have

y = −8

5x + 4.

This shows that the slope of the line is m = −8/5, and they -intercept is b = 4.

Second, to fine the x-intercept, we represent the equation as

y = −8

5(x − 5

2).

Hence, the x-intercept is a = 5/2.

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§P.4 Functions and Their Graphs

The term function was first used by Leibniz in 1673 to denotethe dependence of one quantity on another.

For instance, the area A of a circle depends on the radius raccording to the formula

A = πr2.

We can say that the area (as the dependent variable) is afunction of the radius (as the independent variable).

As a generic function, we often refer to y as the dependentvariable and x as the independent variable. Then to denotethat y is a function of x , we write

y = f (x).

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Definition

A function f on a set D into a set S is a rule that assigns a uniqueelement f (x) in S to each element x in D. We write the functionas

y = f (x).

Remark:

(1) D = D(f ) is the domain of the function f : the set of allpossible input values for the independent variable.

(2) The range of f , R(f ), is the subset of S : the set of allpossible output values for the dependent variable.

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A function can be treated as a kind of machine.

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Graphs of Functions

For any given pair of (x , f (x)), we can find a unique point in thex-y plane to represent it. That is, the function y = f (x) can berepresented by a curve in Cartesian coordinates system.

- x

6y

y = f (x)

b(x , f (x))

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Example 2

Find the domain and range of function:

y = f (x) = 2x2 + 1,

where x ∈ [1, 3].

Solution:

(1) The domain of f is given as

D(f ) = [1, 3].

(2) The range of f isR(f ) = [3, 19].

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Example 3

Find the domain and the range of function:

y = f (x) =√

9− x2 + 1.

Solution:

(1) The domain of f is

D(f ) = [−3, 3].

(2) The range of f isR(f ) = [1, 4].

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Example 4

Find the range of function y =x

x2 + 1, where x ∈ R.

Solution:

For any x ∈ R, we can find y by the relation of

y =x

x2 + 1,

or equivalently, by yx2 − x + y = 0. Now since x is a real number,we have ∆ = 1− 4y2 ≥ 0. This leads to the range of y as

R(y) = [−0.5, 0.5].

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Example 5

Find the range of function y =x

x4 + 1, where x ∈ R.

Solution:

To find the range of y , we need to learn Calculus. Morespecifically, the differentiation in Chapter 2 will help us to solvethis problem. Before starting differentiation, however, we need tofirst introduce the limits and continuity of a function in Chapter1.

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Even and Odd Functions

Definition

Suppose that −x belongs to the domain of f whenever x does.

(1) We say that f is an even function if

f (−x) = f (x) for every x in the domain of f .

(2) We say that f is an odd function if

f (−x) = −f (x) for every x in the domain of f .

Remark:(1) The graph of an even function is symmetric about the y-axis.(2) The graph of an odd function is symmetric about the origin.

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§P.5 Combining Functions to Make New Functions

Functions can be combined in a variety of ways to produce newfunctions.

Like numbers, functions can be added, subtracted, multiplied,and divided (when the denominator is nonzero) to producenew functions.

There is another method, called composition, by which twofunctions can be combined to form a new function.

We can also define new functions by using different formulason different parts of its domain, referred to as “piecewisedefined functions”.

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Sums, Differences, Products, Quotients, and Multiples

Definition

Let f and g be two functions and c is a real number. Then forevery x that belongs to the domains of both f and g , we definefunctions f + g , f − g , f · g , f /g , and cf by the formulas:

(f + g)(x) = f (x) + g(x),

(f − g)(x) = f (x)− g(x),

(fg)(x) = f (x)g(x),(f

g

)(x) =

f (x)

g(x), where g(x) 6= 0,

(cf )(x) = c · f (x).

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Example 6

The functions f and g are defined by the formulas

f (x) =√

x and g(x) =√

2− x .

Find the new functions f + g , f − g , fg , f /g , g/f and f 4 at x .Also specify the domain of each function.

Solution:

(1) (f + g)(x) =√

x +√

2− x , D(f + g) = [0, 2].

(2) (f − g)(x) =√

x −√

2− x , D(f − g) = [0, 2].

(3) (fg)(x) =√

x(2− x), D(fg) = [0, 2].

(4) (f /g)(x) =√

x/(2− x), D(f /g) = [0, 2).

(5) (g/f )(x) =√

(2− x)/x , D(g/f ) = (0, 2].

(6) (f 4)(x) = x2, D(f 4) = [0,∞).

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Example 6

The functions f and g are defined by the formulas

f (x) =√

x and g(x) =√

2− x .

Find the new functions f + g , f − g , fg , f /g , g/f and f 4 at x .Also specify the domain of each function.

Solution:

(1) (f + g)(x) =√

x +√

2− x , D(f + g) = [0, 2].

(2) (f − g)(x) =√

x −√

2− x , D(f − g) = [0, 2].

(3) (fg)(x) =√

x(2− x), D(fg) = [0, 2].

(4) (f /g)(x) =√

x/(2− x), D(f /g) = [0, 2).

(5) (g/f )(x) =√

(2− x)/x , D(g/f ) = (0, 2].

(6) (f 4)(x) = x2, D(f 4) = [0,∞).

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Composite Functions

Definition

Let f and g be two functions. The composite function of f ◦ g isdefined by

f ◦ g(x) = f (g(x)).

The domain of f ◦ g consists of those numbers x in the domain ofg for which g(x) is the domain of f .

Remark:

(1) If the range of g is contained in the domain of f , then thedomain of f ◦ g is just the domain of g .

(2) g is called the inner function and f the outer function. Tocalculate f (g(x)), we first calculate g and then calculate f .

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The output of g becomes the input of f .

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Example 4

Let f (x) = x2 + 1 and g(x) =√

x − 2.

(a) Find the composite function f ◦ g and identify its domain;

(a) Find the composite function g ◦ f and identify its domain.

Solution:

(a) For the first composition, we have

(f ◦ g)(x) = f (√

x − 2) = (√

x − 2)2 + 1 = x − 1.

The domain of f ◦ g is D(f ◦ g) = [2,∞).

(b) For the second composition, we have

(g ◦ f )(x) = g(x2 + 1) =√

x2 − 1.

The domain of g ◦ f is D(g ◦ f ) = (−∞,−1] ∪ [1,∞).

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Example 4

Let f (x) = x2 + 1 and g(x) =√

x − 2.

(a) Find the composite function f ◦ g and identify its domain;

(a) Find the composite function g ◦ f and identify its domain.

Solution:

(a) For the first composition, we have

(f ◦ g)(x) = f (√

x − 2) = (√

x − 2)2 + 1 = x − 1.

The domain of f ◦ g is D(f ◦ g) = [2,∞).

(b) For the second composition, we have

(g ◦ f )(x) = g(x2 + 1) =√

x2 − 1.

The domain of g ◦ f is D(g ◦ f ) = (−∞,−1] ∪ [1,∞).

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Example 4

Let f (x) = x2 + 1 and g(x) =√

x − 2.

(a) Find the composite function f ◦ g and identify its domain;

(a) Find the composite function g ◦ f and identify its domain.

Solution:

(a) For the first composition, we have

(f ◦ g)(x) = f (√

x − 2) = (√

x − 2)2 + 1 = x − 1.

The domain of f ◦ g is D(f ◦ g) = [2,∞).

(b) For the second composition, we have

(g ◦ f )(x) = g(x2 + 1) =√

x2 − 1.

The domain of g ◦ f is D(g ◦ f ) = (−∞,−1] ∪ [1,∞).

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Piecewise Defined Functions

(1) One example is the absolute value function:

|x | =

{x , if x ≥ 0−x , if x < 0.

(2) Another example is the sign function:

sgn(x) =x

|x |=

1, if x > 0−1, if x < 0

undefined, if x = 0.

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§P.6 Polynomials and Rational Functions

Definition

A polynomial is a function P whose value at x is

P(x) = anxn + an−1xn−1 + · · ·+ a1x + a0,

where an, an−1,..., a1 and a0, called the coefficients of thepolynomial, are constants and, if n > 0, then an 6= 0.

Remark: The number n, the degree of the highest power of x inthe polynomial, is called the degree of the polynomial.

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Definition

If P(x) and Q(x) are two polynomials and Q(x) is not the zeropolynomial (i.e., Q(x) ≡ 0), then the function

R(x) =P(x)

Q(x)

is called a rational function.

Remark: By the domain convention, the domain of R(x) consistsof all real numbers x except those for which Q(x) = 0.

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The Quadratic Formula

Theorem

The two solutions of the quadratic equation

Ax2 + Bx + C = 0

where A, B, and C are constants and A 6= 0, are given by

x =−B ±

√B2 − 4AC

2A.

Remark: The quantity ∆ = B2 − 4AC is called the discriminantof the quadratic equation or polynomial.

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Basic Elementary Functions

The following functions are commonly used in Calculus and arethe so-called basic elementary functions:

1) Power functions: xa (a is a real number, witha = n an integer)

2) Exponential functions: ax (a is a positive real number,one important value of a is e = 2.71828)

3) Logarithmic functions: loga(x) (a is a positive realnumber, with loge(x) denoted by ln(x))

4) Trigonometric functions: sin(x), cos(x), tan(x), cot(x),sec(x), csc(x)

5) Inverse Trigonometric functions: arcsin(x), arccos(x),arctan(x), arccot(x), arcsec(x), arccsc(x)

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Elementary Functions

Definition

An elementary function is a function of one variable built from afinite number of basic elementary functions and constants through‘composition’ and ‘combinations’ using the four elementaryoperations (+,−,×, /), where

(f + g)(x) = f (x) + g(x),

(f − g)(x) = f (x)− g(x),

(f · g)(x) = f (x) · g(x),

(f /g)(x) = f (x)/g(x) where g(x) 6= 0.

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Examples of elementary functions include:

f (x) = ln(sin(2x2 + 1)) +√

ex + 1

The polynomial: P(x) = anxn + an−1xn−1 + · · ·+ a1x + a0

The rational function: P(x)/Q(x)

The absolute value function: f (x) = |x |

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Not all functions are elementary. Examples of non-elementaryfunctions include:

The Dirichlet function:

f (x) =

{0 if x is a rational number,1 if x is an irrational number.

The error function:

f (x) =2√π

∫ x

0e−t2dt.

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§P.7 The Trigonometric Functions

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Let C be the circle with center at the origin O and radius 1.Recall that its equation is x2 + y2 = 1.

We use the arc length t as a measure of the size of the angleAOPt . Specifically, the radian measure of angle AOPt is

]AOPt = t radians.

Note that the angle can also be measured in degrees. SincePπ is the point (−1, 0), halfway around the circle C from thepoint A, we have

]AOPπ = π and also ]AOPπ = 180o.

Hence, π = 180o. Some frequently used degrees are, e.g.,π/2 = 90o, π/3 = 60o, π/4 = 45o, and π/6 = 30o.

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Some Useful Identities

(1) The range of Sine and Cosine:

−1 ≤ sin(t) ≤ 1 and − 1 ≤ cos(t) ≤ 1.

(2) The Pythagorean identity:

sin2(t) + cos2(t) = 1.

(3) Periodicity:

sin(t + 2π) = sin(t) and cos(t + 2π) = cos(t).

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(4) Sine is an odd function, and Cosine is an even function:

sin(−t) = − sin(t) and cos(−t) = cos(t).

(5) Complementary angle identities:

sin(π

2− t) = cos(t) and cos(

π

2− t) = sin(t).

(6) Supplementary angle identities:

sin(π − t) = sin(t) and cos(π − t) = − cos(t).

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The graph of sin(x)

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The graph of cos(x)

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Addition Formulas for Sine and Cosine

sin(s + t) = sin(s) cos(t) + cos(s) sin(t),

sin(s − t) = sin(s) cos(t)− cos(s) sin(t),

cos(s + t) = cos(s) cos(t)− sin(s) sin(t),

cos(s − t) = cos(s) cos(t) + sin(s) sin(t).

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Proof: The last formula, cos(s − t) = cos(s) cos(t) + sin(s) sin(t),can be shown by noting that PsPt = Ps−tA. Then by this formula,we can readily derive the remaining three formulas. [The proof isnot required.]

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Letting s = t, by the first and third addition formulas we have

sin(2t) = 2 sin(t) cos(t),

cos(2t) = cos2(t)− sin2(t).

Further, we can show that

sin2(t) =1− cos(2t)

2,

cos2(t) =1 + cos(2t)

2.

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